HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007 EÍDOS 24

2024

1,2,3,4

Mostafa Habibi,

Mohammad Habibi,

Emad Toghroli,

7,8

Maryam Safa,

9,10

Morteza Shariati

Institute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam.

Faculty of Electrical-Electronic Engineering, Duy Tan University, Da Nang 550000, Viet Nam.

Department of Biomaterials, Saveetha Dental College and Hospital, Saveetha Institute of Medical

and Technical Sciences, Chennai, 600 077, India.

Center of Excellence in Design, Robotics, and Automation, Department of Mechanical Engineering, Sharif

University of Technology, Azadi Avenue, P.O. Box 11365-9567, Tehran, Iran. mostafahabibi@duytan.edu.vn.

ORCID: 0000-0003-1638-0338

Department of Materials and Metallurgy, Faculty of Mechanical and Energy Engineering, Shahid Beheshti

University, Tehran, Iran. Mohammad@calut.com.au. ORCID: 0009-0009-0772-5841

Department of Civil Engineering, Calut Company Holding, Melbourne, 800, Australia.

emadtoghroli@calut.com.au. ORCID: 0009-0005-3593-5345

Institute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam.

Faculty of Electrical-Electronic Engineering, Duy Tan University, Da Nang 550000, Viet Nam.

maryamsafa@duytan.edu.vn. ORCID: 0009-0003-0535-7447

Department of Civil Engineering, Calut Company Holding, Melbourne, 800, Australia.

Department of Civil Engineering Discipline, School of Engineering, Monash University, Melbourne 3800,

Australia. mortezashariati@calut.au. ORCID: 0009-0007-0258-0896

Frequency Responses of a Graphene Oxide

Reinforced Concrete Structure

Respuestas de frecuencia de una estructura de hormigón

reforzado con óxido de grafeno

EÍDOS N

Revista Cientíca de Arquitectura y Urbanismo

ISSN: 1390-5007

revistas.ute.edu.ec/index.php/eidos

Abstract:

This paper presents a comprehensive investigation

on the vibrations of reinforced concrete structure by

graphene oxide powders (GOPs) using a polynomial

displacement eld and the Generalized Differential

Quadrature Method (GDQM). The study focuses on

analyzing the dynamic behavior of the structure and

assessing the effects of three different distribution

patterns of GOPs on its vibrations. To accurately model

the deformation of the pressure vessel, a polynomial

displacement eld is employed, taking into account

the complex geometrical and material properties

of the structure. The results highlight the signicant

inuence of the distribution pattern of GOPs on the

natural frequencies of the spherical concrete pressure

vessel. The analysis reveals that variations in the

weight fraction and arrangement of GOPs have a direct

impact on the stiffness and dynamic characteristics

of the structure. Specically, increasing the weight

fraction of GOPs generally leads to higher natural

frequencies, indicating enhanced structural rigidity.

Moreover, the polynomial displacement eld and

GDQM demonstrate their effectiveness in accurately

predicting the vibrations of the reinforced pressure

vessel. The combination of these numerical techniques

enables efcient and reliable analysis of the dynamic

response, allowing for optimization of the design and

performance of spherical concrete pressure vessels.

Keywords: Frequency, vibration; GDQM, graphene

oxide powders, GDQM, stability.

Recepción: 16, 04, 2024 - Aceptación: 15, 05, 2024 - Publicado: 01, 07, 2024

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007

EÍDOS 24

2024

1. INTRODUCTION

Spherical structures nd various appli-

cations across different industries due

to their unique properties and benets.

Here are some notable uses of spherical

structures in different industrial sectors(L.

Lu, Liao, Habibi, Safarpour, & Ali, 2023;

S. Lu, Li, Habibi, & Safarpour, 2023; Ma,

Chen, Habibi, & Albaijan, 2023; Tang, Wu,

Habibi, Safarpour, & Ali, 2023; Y. Wang et

al., 2023). Spherical structures, such as

spherical tanks or spheres, are commonly

used for the storage of liqueed petroleum

gas (LPG) and other volatile uids (Chen &

Lin, 2008; Lee, Yoon, Park, & Yi, 2005). The

spherical shape allows for even distribution

of pressure, resulting in enhanced structur-

al integrity and reduced stress concentra-

tion. Spheres are also used in oil reneries

and petrochemical plants for storing gases

and liquids under high pressure. In addi-

tion, these structures play an important role

in aerospace applications, primarily in the

design of satellites and space exploration

vehicles (Ebrahimi, Hajilak, Habibi, & Safa-

rpour, 2019; Ebrahimi, Mohammadi, Barou-

ti, & Habibi, 2019; Ebrahimi, Supeni, Habi-

bi, & Safarpour, 2020; Habibi, Safarpour,

& Safarpour, 2020; Hashemi et al., 2019;

H Moayedi et al., 2020; Hossein Moaye-

di, Ebrahimi, Habibi, Safarpour, & Foong,

2020; H Moayedi, Habibi, Safarpour, Safa-

rpour, & Foong, 2019; Mohammadgholiha,

Shokrgozar, Habibi, & Safarpour, 2019;

Mohammadi, Lashini, Habibi, & Safarpour,

2019; Oyarhossein et al., 2020; Shariati,

Habibi, Tounsi, Safarpour, & Safa, 2020;

Shariati, Mohammad-Sedighi, Żur, Habi-

bi, & Safa, 2020; Shokrgozar, Safarpour,

& Habibi, 2020). The use of spherical fuel

tanks in spacecraft can ensure uniform fuel

distribution and stability during maneuvers.

In the structure design of the buildings,

spherical structures are used in architec-

ture and construction for their aesthetic ap-

peal and structural advantages. Moreover,

spherical reactors and vessels are utilized

in the chemical and pharmaceutical indus-

try for various processes, such as synthe-

sis, mixing, and containment of reactive

substances. Recently, spherical structures

nd application in renewable energy gen-

eration, such as solar power. Solar con-

centrators, also known as solar spheres,

are spherical reective surfaces used to

concentrate sunlight onto a receiver, which

then converts it into electricity or thermal

energy(Dai, Jiang, Zhang, & Habibi, 2021;

Guo et al., 2021; Kong et al., 2022; Z. Liu,

Su, Xi, & Habibi, 2020; Shao, Zhao, Gao,

& Habibi, 2021; Z. Wang, Yu, Xiao, & Habi-

bi, 2020; Wu & Habibi, 2021; Zhou, Zhao,

Zhang, Fang, & Habibi, 2020). In all these

applications, spherical structures are sub-

jected to the various loadings from static

Resumen:

Este artículo presenta una investigación exhaustiva

sobre las vibraciones de estructuras de hormigón ar-

mado mediante polvos de óxido de grafeno (GOP)

utilizando un campo de desplazamiento polinómico

y el Método de Cuadratura Diferencial Generalizada

(GDQM). El estudio se centra en analizar el comporta-

miento dinámico de la estructura y evaluar los efectos

de tres patrones diferentes de distribución de GOP so-

bre sus vibraciones. Para modelar con precisión la de-

formación del recipiente a presión, se emplea un cam-

po de desplazamiento polinómico, teniendo en cuenta

las complejas propiedades geométricas y materiales

de la estructura. Los resultados resaltan la inuencia

signicativa del patrón de distribución de GOP en las

frecuencias naturales del recipiente a presión de hor-

migón esférico. El análisis revela que las variaciones

en la fracción de peso y la disposición de los GOP tie-

nen un impacto directo en la rigidez y las característi-

cas dinámicas de la estructura. Especícamente, au-

mentar la fracción de peso de los GOP generalmente

conduce a frecuencias naturales más altas, lo que in-

dica una mayor rigidez estructural. Además, el campo

de desplazamiento polinómico y el GDQM demuestran

su ecacia para predecir con precisión las vibraciones

del recipiente a presión reforzado. La combinación de

estas técnicas numéricas permite un análisis eciente

y conable de la respuesta dinámica, lo que permite

optimizar el diseño y el rendimiento de los recipientes

a presión de hormigón esféricos.

Palabras claves: Frecuencia, vibración; GDQM, polvos

de óxido de grafeno, GDQM, estabilidad.

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007 EÍDOS 24

2024

to dynamic loading and, hence, structural

integrity and strength of such structure is of

most important in the design stage.

Although, most of the spherical structures

are made from metals and composite ma-

terials, there can found examples of con-

struction of spherical vessels using con-

crete materials (Hamed, Bradford, & Ian

Gilbert, 2010; Yan, Wang, Zhai, Meng,

& Zhou, 2019; Yue et al., 2022; Zingoni,

2022). Spherical shapes are preferred as

they provide optimal volume-to-surface ra-

tio, allowing for efcient accommodation of

payloads while minimizing weight and heat

transfer. Moreover, concrete are the most

used and available materials of construc-

tion and demonstrates preferable sustain-

ability and load-carrying capacity.

Novel concretes are usually reinforces

with nano-scale materials such carbon

nanotubes (CNTs) (Shahpari, Bamonte, &

Jalali Mosallam, 2022; Siahkouhi, Razaq-

pur, Hoult, Hajmohammadian Baghban,

& Jing, 2021), graphene platelets (GPLs)

(Jaramillo & Kalfat, 2023; Sajjad, Sheikh, &

Hadi, 2022) and graphene oxide powders

(GOPs) (Cong, Cheng, Tang, & Ling, 2023;

Hwangbo et al., 2023; Zeng, Qu, Tian, Hu,

& Li, 2023). These nano-materials demon-

strated desirable improvement in the con-

crete properties.

Structural analyses of the concrete struc-

tures, as a specic type of composite ma-

terials, are widespread. However, one can

categorized these analyses in three major

categories based on the responses of the

structures: static (Godoy, 1987), dynamic

(Dong, Li, & Zheng, 2010), stability (Pan &

Cui, 2010; J. Zhang et al., 2018) analyses.

In the static analysis, the main focus is on

the state of displacement and stress dis-

tribution of the concrete under static and

semi-static loading conditions. In these

analyses, deection of the structure is im-

portant to for functionality of the structure

and stress information is used to design a

superior structure to avoid failure. Mahapa-

tra et al. investigated the large deection

vibration. In the study conducted by Ma-

hapatra et al. (Mahapatra, Kar, & Panda,

2015), the focus was on investigating the

behavior of spherical structures made from

layer-wise composite materials under large

deection vibrations. Their analysis con-

sidered various types of loading, including

thermal and moisture loadings. To math-

ematically model the composite structure,

they employed a higher-order shear de-

formation theory and a general nonlinear

form of Green strain denition. By apply-

ing Hamilton's principle, they derived the

governing equations for the system. The

results revealed the signicant inuence

of the composite pattern and geometry on

the frequency responses of the spherical

structure.

Van Do and Lee (Van Do & Lee, 2020) ex-

tended the research by evaluating both

static deection and free oscillation char-

acteristics of spherical and cylindrical com-

posite shell structures reinforced with Glass

Fiber Reinforced Polymer (GFRP). They uti-

lized an isogeometric analysis methodology

and incorporated different congurations of

reinforcement distribution in the laminate

composite. The analyses employed Red-

dy's shell model and Non-Uniform Ratio-

nal B-Spline (NURBS) curves. The primary

focus of their study was to investigate the

impact of varying reinforcement distribu-

tions on the static and dynamic responses

of curved shell structures.

The post-buckling behavior of spherical

shells is of great importance in numer-

ous industrial applications. Hutchinson

(Hutchinson, 2016) conducted an ana-

lytical investigation to explore the post-

buckling state in spherical structures using

analytical methods. This research aimed to

provide insights into the structural behav-

ior and stability of spherical shells following

the occurrence of buckling.

Additionally, Ghavanloo et al. (Ghavanloo,

Rai-Tabar, & Fazelzadeh, 2019) intro-

duced the nonlocal theory of elasticity in

their analysis of small-size spherical struc-

tures subjected to vibrations. The study

highlighted that the dynamic response of

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007

EÍDOS 24

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small-scale spherical structures differs

from that of macro-scale spheres. The in-

corporation of nonlocal effects allowed for

a more comprehensive understanding of

the vibration behavior in small-sized struc-

tures. Pang et al. (Pang, Li, Chen, & Shan,

2021) examined the effects of boundary

conditions on the vibrational behavior of

a jointed cylindrical and spherical dome

using Rayleigh-Ritz method.

Overall, these studies collectively contrib-

ute to advancing our understanding of the

behavior of spherical structures made from

composite materials under various loading

conditions, including large deection vi-

brations, static deections, post-buckling,

and small-scale dynamic responses.

The study focuses on analyzing the dy-

namic behavior of the pressure vessel and

assessing the effects of three different dis-

tribution patterns of GOPs on its vibrations.

To accurately model the deformation of the

pressure vessel, a polynomial displace-

ment eld is employed, taking into account

the complex geometrical and material prop-

erties of the structure. The results highlight

the signicant inuence of the distribution

pattern of GOPs on the natural frequencies

of the spherical concrete pressure vessel.

The analysis reveals that variations in the

weight fraction and arrangement of GOPs

have a direct impact on the stiffness and

dynamic characteristics of the structure.

2. METHODOLOGY

2.1. GOP Distribution in Thickness

Figure 1 illustrates an FG-GOPRC spheri-

cal structure along with different patterns

depicting the distribution of GOPs (Gra-

phene Oxide Polymer Reinforced Compos-

ites) in the thickness direction of the spheri-

cal shell.

Moving forward, the material properties rel-

evant to this investigation were determined

using the Halpin-Tsai homogenization

method. This method allows for the estima-

tion of the effective material properties of

composite materials, such as the Young's

modulus. By employing the Halpin-Tsai

approach, we are able to calculate the

Young's modulus, which is a key param-

eter for characterizing the stiffness of the

material (Z. Zhang et al., 2020) as follows:

(1)

in which,

and

Figure 1. Schematic view of various GPL distribution patterns

a)X-GOP b)O-GOP c)UD-GOP

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In addition to the Young's modulus, the

density and Poisson's ratio also consid-

ered as follows:

(2)

And the shear modulus of the system can

be dened as:

(3)

Lastly, the different pattern of GOPs can be

dened as followings for GOP–X (Eq. (4a)),

GOP–O (Eq. (4b)) and GOP–UD (Eq. (4c)),

(Z. Zhang et al., 2020):

(4a)

(4b)

(4c)

Where

changes from 1 layer to

2.2. Equation of Motion

The displacement eld of the spherical

shell is expressed by

(5a)

(5b)

(5c)

Where strain displacement can be dened

as follows

(6)

By substituting

=rØ sin

in Eq. (6)

(7)

The unknown quantities in the above equa-

tions could be found in (Al-Furjan et al.,

2022; Al-Furjan, Oyarhossein, Habibi, Sa-

farpour, & Jung, 2020; Habibi, Moham-

madi, Safarpour, Shavalipour, & Ghadiri,

2021; H. Moayedi et al., 2020)

2.3. Hamilton’s Principle

The equations that determine motion are

derived based on Hamilton’s principle:

(8)

Where

, and

stand for the kinetic

energy, potential energy, and work done by

the system respectively. The kinetic energy

of the moving plate is indicated as follows

(9)

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EÍDOS 24

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The potential energy of the axially moving

plate is illustrated as follows

(10)

The work done by the system:

(11)

Where

indicates the In-plane mechani-

cal loading. Substituting Eqs. (9), (10), and

(11) into Eq. (8), the governing equations

of motion and boundary conditions are

obtained.

3. SOLUTION PROCEDURE

To illustrate the approximations involved in

the Harmonic Differential Quadrature Method

(HDQM) using a one-dimensional function,

the following equation expresses the

derivative of F(Ø) in terms of Ø:

(12)

here

indicates the total number of dis-

crete grid nodes selected through the solu-

tion domain.

The equation establishes a relationship be-

tween the pth derivative of the function F

with respect to Ø and the variable Ø itself.

This relation serves to demonstrate how

the HDQM method approximates the de-

rivatives of a function using a discrete set

of values and their corresponding weights.

By employing this approach, the HDQM

method allows for the efcient and accu-

rate calculation of derivatives in numeri-

cal analysis and computational modeling.

The term

(p)

shows the weight coefcients

(

j=1,2,…,N_∅

) at the

th grid-point located

in the solution domain. The weight coef-

cients related to the rst-order derivatives

(1)

for

i≠j

would be determined through

the following relation:

(13)

here

(14)

The weight coefcients related to the rst-

order derivatives G_ij^((1) ) when i=j can

be acquired as below

(15)

The weight coefcients related to the

second-order derivatives

(2)

when

i≠j

would be acquired through the subsequent

relation

(16)

The weight coefcients related to the sec-

ond-order derivatives

(2)

when

i=j

would

be determined as

(17)

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Also, the Chebyshev–Gauss–Lobatto grid

distribution is chosen. In this distribution,

the co-ordinates of grid points (

∅_i,Θ_j

) are

calculated by the owing equation across

the reference surface.

The displacement eld expressions are

given as below,

(18)

Using the above equations, the following

linear set of equations are obtained:

(19)

By solving Eq. (19), the natural frequency

of the system can be achieved.

4. NUMERICAL RESULTS AND

DISCUSSION

4.1. Materiel Properties

The material properties associated with

GOPs reinforcement as well as concrete

matrix, and spherical vessel are presented

in Table 1 adopted from Ref. (Z. Zhang et al.,

2020).

Table 1. The properties of GOPs, polymer, and

pressure vessel

Polymer

epoxy(matrix)

Pressure

vessel

GOPs

=0.42

(kg)=260

GOP

=0.165

(Mpa)=25

∅

=10 [deg]

GOP

(kg/

m³)=1090

∅

=170 [deg]

GOP

(

Gpa

)=444.8

GOP

(nm)=500

GOP

(nm)=0.95

In addition the following dimensionless pa-

rameters are dened for natural frequency

Ω and internal pressure P



4.2. Validation

The new material and method presented

in the current study need to be veried by

comparing to the results of other studies

in the similar structure and loading condi-

tions. In this regard, the problem of vibra-

tional behavior evaluation from Liu et al.

(D. Liu, Zhou, & Zhu, 2021). The results

are presented for two different parameter

variation namely the mode number of vi-

bration and the GOP distribution patterns.

It is seen that the current method provides

similar results to the selected reference

which validate our methodology. On the

other hand, different variation in pattern of

GOP results in different natural frequency

of pressure vessel shell structure which

will be discussed in next sections in details

along with several other parameters.

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4.3. Parametric Study

This section examines the impact of dif-

ferent parameters and congurations on a

nanocomposite shell of a pressure vessel.

The number of layers within the nanocom-

posite structure plays a signicant role in

determining the natural frequency of the

pressure vessel, as depicted in Figure 2.

The gure illustrates that an increase in the

number of layers, denoted as N

, within

the GOP-X conguration leads to a corre-

sponding increase in the natural frequen-

cy of the structure, particularly for a low

number of layers. However, for a higher

number of layers (N

> 50), an increase in

the number of layers has minimal effect on

the natural frequency, assuming all other

parameters remain constant. In contrast,

an increase in the number of layers in the

GOP-O conguration has the opposite ef-

fect on the natural frequency of the pres-

sure vessel structures. Specically, as the

number of layers in GOP-O increases, the

natural frequency decreases. On the other

hand, altering the number of layers in the

uniform conguration does not have any

inuence on the natural frequency of the

structure. Therefore, this analysis demon-

strates the signicant inuence of the num-

ber of layers and different congurations

on the natural frequency of nanocompos-

ite shell structures, with distinct behaviors

observed for different congurations.

Table 3. Effects of number of layers and

distribution pattern of GOP

on the dimensonless frequency

of concrete pressure vessel

GOP-X GOP-U GOP-O

7 0.6309 0.6309 0.6310

9 0.6311 0.6309 0.6308

11 0.6312 0.6309 0.6307

13 0.6312 0.6309 0.6307

15 0.6312 0.6309 0.6307

25 0.6313 0.6309 0.6306

35 0.6313 0.6309 0.6306

55 0.6313 0.6309 0.6306

100 0.6313 0.6309 0.6306

The natural frequency of the shell is inu-

enced by an increase in the thickness of

the structure (h) in comparison to the radi-

us of the pressure vessel (r), as illustrated

in Figure 2. This analysis considers various

weight fractions of GOP (W

GOP

). The gure

demonstrates that an increase in the thick-

ness of the pressure vessel shell leads

to a corresponding increase in the natu-

ral frequency of the structure, regardless

of the weight fraction of GOP. This effect

is expected, as increasing the thickness

while keeping the radius constant results

in a higher stiffness within the structure,

ultimately raising its natural frequency. Fur-

thermore, an increase in the weight frac-

tion of GOP contributes to an increase in

the stiffness of the structure, consequently

Table 2. Frequency parameter of the composite spherical shell

Mode

number

Epoxy GPL-UD GPL-X GPL-O

Ref. (D.

Liu et al.,

2021)

Present

Ref. (D.

Liu et al.,

2021)

Present

Ref. (D.

Liu et al.,

2021)

Present

Ref. (D.

Liu et al.,

2021)

Present

1 19.2432 19.2361 40.0892 40.0045 29.6019 29.5891 43.7133 43.6901

2 1.1613 1.1601 2.4194 2.4012 2.4203 2.4117 2.4196 2.4032

3 1.8362 1.8351 3.8253 3.8212 3.8263 3.8215 3.8251 3.8182

4 2.4635 2.4602 5.1322 5.1310 5.1324 5.1301 5.1309 5.1292

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elevating the natural frequency. This rela-

tionship is consistent with the expectation

that higher weight fractions of GOP en-

hance the overall stiffness of the nanocom-

posite shell. Therefore, both an increase

in the thickness of the structure relative to

the pressure vessel radius and an increase

in the weight fraction of GOP lead to an

increase in the natural frequency of the

structure. These ndings align with the an-

ticipated effects of stiffness enhancement

resulting from changes in the structural

parameters and the incorporation of GOP

within the nanocomposite shell.

Figure 2. Effect of h/r aspect ratio and W

GOP

on the natural

frequency of pressure vessel

The natural frequency of the entire shell

model is also inuenced by the total mass

of the spherical vessel, as shown in Figure

3 for various thicknesses. An increase in the

mass of the pressure vessel, while keep-

ing the geometrical parameters constant,

corresponds to a decrease in the material

density. This decrease in density results

in a reduction in the natural frequency of

the structure, as can be observed from the

curves in Figure 4. Similarly, the effect of

the h/r aspect ratio (the ratio of thickness

to radius) on the natural frequency follows

a similar trend to that depicted in Figure 2.

As the h/r aspect ratio increases, there is a

corresponding increase in the natural fre-

quency of the shell structure.

Therefore, the total mass of the spherical

vessel and the h/r aspect ratio play signi-

cant roles in determining the natural fre-

quency of the entire shell model. Increas-

ing the mass, which decreases the material

density, results in a decrease in the natural

frequency. Additionally, an increase in the

h/r aspect ratio leads to an increase in the

natural frequency, highlighting the inu-

ence of structural dimensions on the dy-

namic characteristics of the shell.

Figure 3. Effect of total mass of the pressure vessel m

and

h/r on the natural frequency of pressure vessel

An increase in the angle

generally leads

to a decrease in the natural frequency of

the structure for all thickness values, as il-

lustrated in Figure 4. However, the extent of

this decrease varies depending on the spe-

cic thickness and the value of the angle

In cases with low angles, it is observed that

the natural frequency experiences a more

signicant increase compared to other an-

gle values. The gradient of increase in the

natural frequency is higher for lower angle

values. This implies that small changes in

the angle

at lower angles have a more

pronounced effect on reducing the natu-

ral frequency of the structure. Conversely,

as the angle

increases, the decrease in

the natural frequency becomes less pro-

nounced. The inuence of the angle

the natural frequency is still present, but

the effect is relatively milder compared to

lower angle values.

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Figure 4. Effect of angle

and h/r on the natural frequency

of pressure vessel

The boundary condition plays a crucial role

in determining the natural frequency of the

structure, as shown in Table 4. By altering

the boundary condition, the natural frequen-

cy of the structure varies across different

values of the h/r (thickness to radius) ratio.

Among the different boundary conditions

considered, the simply supported bound-

ary condition results in the smallest natural

frequency for a constant value of the h/r ra-

tio. This indicates that the structure exhibits

a lower frequency response when it is sup-

ported at the ends, allowing for some de-

gree of freedom in terms of displacement.

On the other hand, the clamped bound-

ary condition corresponds to the highest

frequency values. This boundary condition

restricts the structure from experiencing

any displacement at the ends, leading to a

higher stiffness and consequently a higher

natural frequency.

These ndings demonstrate that the state

of the boundary condition is a determining

factor in calculating the natural frequency of

the structure. Different boundary conditions

result in varying natural frequency values,

with the simply supported condition yield-

ing the lowest frequency and the clamped

condition producing the highest frequency.

In the previous parametric studies, it was

assumed that the internal pressure of the

pressure vessel is equal to the ambient

pressure. However, in reality, the pres-

sure inside the pressure vessel is positive.

The variation in the internal pressure of the

pressure vessel has a signicant impact on

the natural frequency of the structure, as

illustrated in Figure 5.

Table 4. Effects of \h/r and type of boundary

condition on the dimensonless frequency of

concrete pressure vessel

h/r S-S C-S C-C

0 0.55942 0.57244 0.59619

0.05 0.57459 0.58809 0.61280

0.1 0.58263 0.59646 0.62015

0.15 0.58848 0.60291 0.62601

0.2 0.59315 0.60840 0.63128

The gure reveals that changing the pa-

rameter P



from positive values to negative

values substantially alters the behavior of

the frequency curves. When the parameter



has positive or zero values, the natural

frequency monotonically increases for all

values of the h/r ratio. This implies that as

the internal pressure increases, the natural

frequency of the structure also increases.

However, in the case of negative values of

parameter P



, the behavior of the natural

frequency changes. For lower values of the

h/r ratio, the natural frequency decreases

as the h/r ratio increases. This implies that

the internal pressure has a counterintuitive

effect on the natural frequency for certain

values of the h/r ratio.

In each value of parameter P



, there is a

turning point in the curves where the natu-

ral frequency begins to rise. This indicates

that there is a critical value of the h/r ratio at

which the inuence of the internal pressure

on the natural frequency changes. Interest-

ingly, all the curves converge and coincide

at a high value of the h/r ratio. This sug-

gests that at sufciently large values of the

h/r ratio, the inuence of the internal pres-

sure becomes negligible, and the natural

frequency reaches a steady state. The vari-

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007 EÍDOS 24

2024

ation in the internal pressure of the pres-

sure vessel has a profound impact on the

natural frequency of the structure. Positive

and zero values of the internal pressure

lead to a monotonic increase in the natural

frequency, while negative values introduce

complex behavior, with a turning point and

counterintuitive effects for certain h/r ratios.

The curves ultimately converge and coin-

cide at high h/r ratios, indicating the dimin-

ishing inuence of the internal pressure.

Figure 5. Effect of parameter P and h/r on the natural

frequency of pressure vessel

Figure 6 illustrates the simultaneous effects

of the h/r aspect ratio and the weight frac-

tion of GOP on the natural frequency of the

structure. Similar to Figure 2, an increase

in both of these parameters leads to an in-

crease in the natural frequency of the struc-

ture. However, one notable feature of the

present graph is the semi-linear change in

the natural frequency due to variations in

the weight fraction of the GOP, which was

not apparent from Figure 2.

The graph demonstrates that as the h/r

aspect ratio increases, there is a corre-

sponding increase in the natural frequency

for various weight fractions of GOP. This

relationship is consistent with the under-

standing that a larger h/r ratio results in a

stiffer structure, leading to higher natural

frequencies.

Furthermore, the weight fraction of the

GOP has a distinct effect on the natural

frequency. In Figure 6, it is observed that

as the weight fraction of GOP increases,

the natural frequency also increases.

However, unlike in Figure 2, where the

relationship between weight fraction and

natural frequency appeared to be linear,

Figure 6 reveals a semi-linear change in

the natural frequency. This suggests that

the inuence of the weight fraction of GOP

on the natural frequency is more nuanced

and exhibits a gradual change as the

weight fraction varies.

Figure 6. Effect of W

GOP

and h/r on the natural frequency of

pressure vessel

In Figure 7, the linear increase of the natu-

ral frequency with an increase in the weight

fraction of GOP is observed, similar to the

previous ndings. This indicates that as

the weight fraction of GOP increases, the

natural frequency of the structure also in-

creases in a linear fashion. Additionally,

the effect of pressure vessel mass on the

natural frequency is evident in Figure 7.

Higher values of pressure vessel mass

correspond to lower values of the natural

frequency. This means that as the mass of

the pressure vessel increases, the natural

frequency of the structure decreases. The

relationship between pressure vessel mass

and natural frequency can be attributed

to the increased inertia of the structure

caused by the higher mass. The higher

mass contributes to a greater resistance

against displacement and thus results in a

lower natural frequency.

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007

EÍDOS 24

2024

Figure 7. Effect of total mass of the pressure vessel m

and

GOP

on the natural frequency of pressure vessel

Figure 8 illustrates the simultaneous ef-

fects of the angle

and the weight frac-

tion of GOP (Graphene Oxide Polymer) on

the natural frequency of the structure. As

observed in Figure 2, an increase in the

weight fraction (W

GOP

) leads to an increase

in the natural frequency of the structure.

Figure 8. Effect of angle

and W

GOP

on the natural

frequency of pressure vessel

However, an important feature depicted in

Figure 8 is the decrease in the natural fre-

quency of the structure with an increase

in the angle

from 140° to 170°. This ob-

servation contrasts with the general trend

seen in the previous gures. The decrease

in the natural frequency with an increase

in the angle

suggests that certain angu-

lar congurations have a dampening ef-

fect on the dynamic response of the struc-

ture. This behavior may be attributed to

the specic geometry and material prop-

erties associated with the angle

range

of 140° to 170°. It is worth noting that this

unique feature was not evident in Figure

2, indicating that the interaction between

the angle

and the weight fraction of GOP

introduces additional complexities to the

natural frequency response.

5. CONCLUSION

This paper presents a comprehensive

investigation on the vibrations of spheri-

cal concrete pressure vessels reinforced

by graphene oxide powders (GOPs) us-

ing a polynomial displacement eld and

the Generalized Differential Quadrature

Method (GDQM). This research contrib-

utes to advancing the understanding of the

vibrational behavior of graphene oxide-re-

inforced spherical concrete pressure ves-

sels, particularly focusing on the inuence

of different distribution patterns of GOPs.

The ndings offer valuable insights for en-

gineers and researchers involved in the

design and analysis of pressure vessels,

contributing to improved structural integrity

and performance. The main results could

be given in the following items:

• Firstly, it is observed that higher pres-

sure vessel masses result in lower

natural frequencies. This is due to the

increased mass, which contributes to

higher inertia and stiffness, thus re-

ducing the natural frequency of the

structure.

• Secondly, variations in the internal

pressure of the pressure vessel have

a signicant inuence on its natural

frequency. Changes in internal pres-

sure directly affect the structural dy-

namics, leading to variations in the

natural frequency.

• Additionally, an increase in the thick-

ness of the pressure vessel shell is

found to cause an increase in the

natural frequency. This is attributed

to the higher stiffness resulting from

HABIBI, et al. - Frequency Responses of a Graphene Oxide Reinforced Concrete Structure. pp. 63-79 ISSN:1390-5007 EÍDOS 24

2024

the thicker shell, leading to a higher

natural frequency of vibration.

• Furthermore, the weight fraction of

GO P demonstrates a linear rela-

tionship with the natural frequency.

Increasing the weight fraction pro-

portionally increases the natural fre-

quency, indicating that the addition

of GO P enhances the overall stiff-

ness and dynamic characteristics of

the pressure vessel.

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